My American Mathematical Society Feature Column Articles
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York
The Public Awareness Office of the American Mathematical Society sponsors a Feature Column which consists of expository web based columns on a wide variety of topics. For three years I wrote a monthly Feature Column and now I share writing these columns with David Austin, William Casselman, and Anthony Philips.
Listed below, in reverse chronological order, are the columns I wrote with brief "abstracts" for their content. A direct link to each column is provided: Most of the columns deal with geometry, combinatorics, game theory and fairness questions. In a general way, I like each column to show the two faces of mathematics: its appealing theory and its wide applicability.
Are Precise Definitions a Good Idea?
What properties of a geometric figure make it a "polygon?" It is hard to pin down a definition of the term "polygon" which includes all of the geometric figures that now would be called polygons. Sometimes a "flexible" approach to definitions allows one the room for ideas that were born to deal with something relatively narrow to grow and evolve. Sometimes precise definitions encourage memorization at the expense of conceptual understanding.
Mathematics and Ecology
Another in a series of columns about how mathematics is used in scholarly areas outside of mathematics. In this column some comments are made about the role of taxonomy in ecology and also there is a discussion of rarefaction. This has to do with how to measure the diversity of the Earth's animals and plants.
This was a Mathematics Awareness Month column. It gives some biographical information about men and women who became mathematicians and who "studied" mathematics as well as others who had careers as mathematicans but "studied" other academic subjects to some extent.
Mathematics and Psychology
This is the second in a series of columns devoted to the way mathematics enriches subjects outside of mathematics. This column deals with the way mathematics has helped illuminate many aspects of psychology. The emphasis is on how measurement works in physics versus what happens in the social sciences, n particular, psychology.
Mathematics and Chemistry: Partners in a Changing World
This is the first in a series of columns devoted to the way mathematics enriches subjects outside of mathematics. This column deals with the way mathematics has helped illuminate many aspects of chemistry. The emphasis is on graph theory ideas used in chemistry.
Magical Mathematics - A Tribute to Martin Gardner
Martin Gardner, though not a professional mathematician, has had a tremendous effect on popularizing and promoting mathematics. His Scientific American columns and his many books based on these columns have inspired many people to become mathematicians and to see the interest and value of the subject.
Words and More Words
Hurricane Sandy Meets Mathematics
More Precious than Gold?
Going, Going, ..., Gone!
Many people think computers can solve any problem that humans set them to work on. Not only are there problems that can be solved on computers but there are easy to state problems which can be solved even for moderate sizes of the problems - finding an optimal route for a traveling salesman. This article surveys some mathematical insights into computational complexity.
Price of Anarchy
In many situations people do what seems to best action for them. However, game theory shows that when all the players in some games follow their "best choice" that everyone winds up in a place where they are not so well off. An example of this kind of thing is congestion in road systems. The price of anarchy is an attempt to measure how far from the optimum which might attainable with "regulation" (telling people what route to drive on rather than letting them select for themselves) when all people in a "game" act on their own.
Mathematics and Sports
Keep on trucking
A survey of ideas about vehicle routing problems and the related Traveling Salesman problem is provided.
Matching markets aim to create "partnerships" of various kinds not based on price but on preferences. Gale-Shapley found a remarkable algorithm, the deferred acceptance algorithm which finds two (sometimes only one) "stable" pairings. Recently, these ideas have been successfully extended to pairing children with K-12 public schools in large urban areas.
Mathematics and Climate
This column surveys some ideas about how mathematical models are constructed using differential equations, and discusses a bit of what makes climate modeling especially difficult. There is a discussion of the role of albedo in helping measure how much energy from the sun gets absorbed by the Earth how much gets reflected back into space.
People making a difference
This column is a tribute to the great American geometry Victor Klee. HIs work on linear programming and what has come to be called the art gallery theorem is discussed.
Suppose one has a related collection of objects (typically binary sequences or "words" in some alphabet). There is a way of measuring how far two of the objects are from each other. The goal is to construct a graph where two of the objects (thought of as vertices of the graph) are joined by an edge when the distance between these two is small. The goal is to find a tour of the graph which visits each vertex once and only once - a Hamiltonian circuit. Such a circuit in this case is known as a Gray code, named for Frank Gray who found such codes for the binary sequences of length n.
The process of electing a president
This column surveys mathematical insights into the process of conducting elections, as well as issues related to the fairness of election decision methods.
This column surveys some of the ways that geometrical ideas can be used to get insight into urban problems like collecting garbage or removing snow efficiently; routing school buses and meals on wheels vans, etc. The tool involved is basic graph theory and the graph theory terms explored are Eulerian circuits, Hamiltonian circuits, etc.
We are all familiar with the fact that time it takes a taxi many cities seems to take longer than we expect, even in light traffic. Perhaps one reason for this is that humans are so accustomed to think in terms of Euclidean distance that we forget that in a "grid" like city a more natural distance is what has come to be called taxicab distance, where one can't move "diagonally" but only in the directions of the grid. This column surveys some of the amazing issues that studying taxicab distance leads to.
Mathematics and the brain
Rationality and game theory
Mathematics and internet security
Trees: A mathematical tool for all seasons
Sales and chips
Mathematics and cosmology
Resolving bankruptcy claims
Euler's polyhedral formula II
Euler's polyhedral formula
Voting games II
Voting games I
Bin packing and machine scheduling
Colorful mathematics IV
Colorful mathematics III
Colorful mathematics II
This column continues the study of a variety of coloring problems.
Colorful mathematics I
Problems that assign labels or colors to the dots and lines of a graph have many applications. One of mathematics' most colorful theorems is the Four Color Theorem which asserts that any graph which can be drawn in the plane can be vertex colored with four or fewer colors.
A discrete mathematical gem
James Joseph Sylvester raised the question of what can be said about a set S of points in the plane with the property that for any two points P and Q of S there is another point of S on the line PQ? Can you prove that S consists of a set of collinear points? This result is now known as the Sylvester-Gallai problem and it has led to many interesting problems in discrete geometry.
The number 12 can be written as the product of 3 and 4, but the only way to write 3 as a product is as 3x1. The number 4 can be "broken down" further as 2x2. The primes are positive integers p (greater than or equal to 2) where the only way to write p as a product is 1xp or px1. This column surveys some basic properties of primes and how the notation of a prime can lead to applications outside of mathematics.
Oriented matroids: The power of unification
Matroids involve trying to "generalize" the notation of things being "independent" of each other that arise when studying vectors or circuits in graphs. Oriented matroids are an attempt to "unify" ideas that have arisen in a variety of settings. This column discusses how ideas in different parts of combinatorics and geometry are unified using the idea of an oriented matroid.
Mathematics and art
Art has inspired new mathematics and mathematics has inspired artists. This column surveys examples including a discussion of M.C. Escher.
Combinatorial games II: Different moves for left and right
The December, 2002 column dealt with impartial games, ones where there weren't special "pieces" for each player to move like in Nim. Here partisan games are considered, such as John Horton Conway's Hackenbush. Conway showed deep connections between "values" for games and the numbers now known as the surreal numbers.
Digital revolution III -Error Correction codes
Richard Hamming discovered a way of taking a binary sequence, adding additional characters to the sequence, which enable the new binary sequence to correct bits in the sequence that might get changed due to random noise, when the sequence was transmitted from one person to another. This work is related to that of Claude Shannon on information theory. This work has made possible many new technologies including DVD's and cell phones.
Matroids: The value of abstraction
One reason that many people find mathematics difficult is that they view it is as being so abstract. Sometimes it is abstraction get a richer view of mathematics because one is able to see that some phenomenon which occurs in many examples, can be abstracted so that all of the examples can be thought of special cases of the more abstract situation. Matroids are way of abstracting a variety of examples that occur in graph theory, linear algebra, and other parts of mathematics.
Combinatorial games I - The world of piles of stones
Combinatorial games is the branch of mathematics devoted to getting insight into games that don't involve chance, such as checkers or chess. An interesting type of game of this kind is where the "board" has no special pieces for each player. The classic example is Nim, where there are piles of stones and a player makes a move by selecting one of the piles and takes some subset of the stones or all of them. Remarkably, there is a sense in which all such combinatorial games are like Nim.
Linkages II - Old wine in new bottles
This column continues the September, 2002 column and treats other aspects of linkages, sticks that are pinned to each other at their ends.
Digital revolution II - Data compression codes and technologies
David Huffman pioneered the use of binary codes which enabled one to take a string of zeros and ones and replace it with a cleverly constructed shorter string which made it possible to recover the original binary string. What have come to be called Huffman codes and other data compression techniques make it possible to compress text, images, and video. Many new technologies, including smart phones, employee such schemes.
Linkages: From fingers to robot arms I
Linkages were already being studied in the the 19th century by Alfred Kempe (of 4-color problem fame) but for applied and theoretical reasons. A linkage is a collection of "sticks" that are pinned together in someway and one is typically interested in the motion of a point or endpoint on one of the sticks. Linkages are still of great theoretical and applied interest, in particular, in robotics.
The digital revolution I: Barcodes
The digital revolution refers to the digitally based technologies most notably the stored program digital computer. In conjunction with the digital revolution has been the development of codes for the purpose of hiding, compressing, tracking, compressing, synchronizing, etc. Bar codes were developed in part for the purpose of enabling the speeding of the inventory, distribution and sales of merchandise. One of the early use of barcodes was to speed the delivery of mail. This column explores some of the mathematics related to barcodes.
This column continues the survey of apportionment methods begun in the May, 2002 column but can more or less be read independently of the first column.
In the US, states are to be assigned seats in the House of Representatives based on the population sizes of the states, while in European democracies parties are assigned sets in "parliament" based on the votes the parties gets. These two situations don't lead to identical treatment mathematically because in the US each state must get at least one seat. The historical background for the way mathematics has been used to solve "apportionment" problems is surveyed.
Mathematics and the genome
Genetics, which benefited greatly from mathematics, changed radically after the introduction of the Crick-Watson model. However, many has proved equally useful here, in finding "distances" between genes in different specials, phylogenetic trees, and attempts to locate genetically "active" stretches of DNA.
Voting and Elections
Nearly all elections in America are conducted using a ballot where one chooses only one candidate and the winner is based on which candidate gets the largest number of votes. However, many other kinds of ballots could be used (ordinal ballots, cardinal ballots, approval ballots, etc.) and using the greater information available from those ballots many different approaches to choosing a winner are available. Kenneth Arrow's result about election decision methods is discussed.
Nets: A tool for representing polyhedra in two dimensions
A net is a plane polygon (with indicated fold lines) which folds to a polyhedron. It is still an open question whether or not there is a way to cut some of the edges of a convex polyhedron to obtain a net (Shephard's Conjecture).